A Non-Oscillatory Eulerian Approach to Interfaces

in Multimaterial Flows (The Ghost Fluid Method)

Ronald P. Fedkiw �

Tariq Aslam y

Barry Merriman *

Stanley Osher *

October 19, 1998

Abstract

While Eulerian schemes work well on most gas ows, they have been

shown to admit nonphysical oscillations near some material interfaces,

especially in multimaterial problems associated with deformable solids.

In contrast, Lagrangian schemes work well at multimaterial interfaces,

but su�er from their own di�culties in problems with large deforma-

tions characteristic of most gas ows. We believe that the most robust

schemes will combine the best properties of Eulerian and Lagrangian

schemes. In fact, this paper is dedicated to developing an Eulerian

scheme which treats the interface in Lagrangian way, i.e. as a Heavi-

side function with no smearing along the lines of [10], [4] and [3].

In this paper, we use a level set function [24, 30, 25] to track the

motion of a multimaterial interface in an Eulerian framework. In addi-

tion, the use of ghost cells (actually ghost nodes in our �nite di�erence

framework) and a new Isobaric Fix [6] technique allows us to keep

the density pro�le from smearing out, while still keeping the scheme

robust and easy to program along the lines of [29] with simple exten-

sions to multidimensions and multilevel time integration, e.g. Runge

�UCLA - Research supported in part by ONR N00014-97-1-0027 and ONR N00014-97-

1-0968yLos Alamos National Laboratory - performed under the auspices of the U.S. Depart-

ment of Energy

1

Kutta methods. In contrast, [3] and [4] all use ill-advised dimensional

splitting for multidimensional problems and [10], [3], and [4] all su�er

from great complexity when used in conjunction with multilevel time

integrators.

2

1 Introduction

Eulerian schemes work well for most problems and can accurately and e�-ciently handle large deformations characteristic of gases. However, they canadmit nonphysical oscillations near material interfaces due to the smearedout density pro�le and the radical change in equation of state across a ma-terial interface. Lagrangian schemes work well on material interfaces, sincethey do not smear out the density pro�le and it is clear which equation ofstate is valid at each point. Unfortunately, Lagrangian schemes have theirown problems when subjected to large deformations such as those charac-teristic of gas ow. For a good summary of both Eulerian and Lagrangianschemes, see [2].

Our method consists of combining the robustness of an Eulerian schemewith a multimaterial interface method characteristic of a Lagrangian scheme.We do this by tracking the interface with a level set function [24, 30] whichgives the exact subcell interface location. At this interface, we solve an ap-proximate Riemann problem similar to the methods in [10], [4] and [3]. In[10], [4] and [3] the authors use schemes that are intricate in one dimensionand can only be extended to multiple dimensions with dimensional splittingin time. In addition, multilevel time integrators, such as the Runge Kuttamethods, are hard to implement for these methods. In contrast, our methoddraws on ideas from [29] which enables us to treat multidimensional calcula-tions without time splitting and allows the easy and e�cient implementationof Runge Kutta methods. This is done with an elegant use of ghost cells andthe application of a new Isobaric Fix technique [6].

We make note of an alternative method of solving interface problemswith Eulerian schemes. In [17], [14], and [5] the authors allow the errors indensity associated with a smeared out material interface, and they attemptto �x these errors by modifying the internal energy to get exact cancelation ofthese error. While some of the preliminary results with a gamma law gas, seee.g. [31], are extremely promising, it is unclear that it will always be possibleto remedy the errors associated with a smeared out density pro�le. In fact,the general pressure evolution equation [5] has a discontinuous coe�cientwith no meaningful regularization for general equations of state. We havepushed this equation to its limits in [21] and have been disappointed by itslack of robustness. In general, we advocate schemes which do not smear out

3

the density pro�le.

4

2 Equations

2.1 Euler Equations

The basic equations for two-dimensional compressible ow are the 2D Eulerequations, 0

BBB@�

�u

�v

E

1CCCAt

+

0BBB@

�u

�u2 + p

�uv

(E + p)u

1CCCAx

+

0BBB@

�v

�uv

�v2 + p

(E + p)v

1CCCAy

= 0 (1)

where t is time, x and y are the spatial dimensions, � is the density, u andv are the velocities, E is the total energy per unit volume, and p is thepressure. The total energy is the sum of the internal energy and the kineticenergy,

E = �e+�(u2 + v2)

2(2)

where e is the internal energy per unit mass. The one-dimensional Eulerequations are obtained by setting v = 0.

In general, the pressure can be written as a function of density andinternal energy, p = p(�; e), or as a function of density and temperature,p = p(�; T ). In order to complete the model, we need an expression for theinternal energy per unit mass. Since e = e(�; T ) we write

de =

�@e

@�

�T

d� +

�@e

@T

��

dT (3)

which can be shown to be equivalent to

de =

�p� TpT

�2

�d� + cvdT (4)

where cv is the speci�c heat at constant volume. [1]The sound speeds associated with the equations depend on the partial

derivatives of the pressure, either p� and pe or p� and pT , where the change

5

of variables from density and internal energy to density and temperature isgoverned by the following relations

p� ! p� ��p� TpTcv�2

�pT (5)

pe ! p� +

�1

cv

�pT (6)

and the sound speed c is given by

c =

rp� +

ppe�2

(7)

for the case where p = p(�; e) and

c =

sp� +

T (pT )2

cv�2(8)

for the case where p = p(�; T ).The eigenvalues and eigenvectors for the Jacobian matrix of ~F (~U) are

obtained by setting A = 1 and B = 0 in the following formulas, while thosefor the Jacobian of ~G(~U) are obtained with A = 0 and B = 1.

The eigenvalues are

�1 = u� c; �2 = �3 = u; �4 = u+ c; (9)

and the eigenvectors are

~L1 =

�b22+

u

2c;�b1u

2� A

2c;�b1v

2� B

2c;b12

�; (10)

~L2 = (1� b2; b1u; b1v;�b1) ; (11)

~L3 = (v; B;�A; 0) ; (12)

~L4 =

�b22� u

2c;�b1u

2+A

2c;�b1v

2+B

2c;b12

�; (13)

6

~R1 =

0BBB@

1u�Ac

v �Bc

H � uc

1CCCA ; ~R2 =

0BBB@

1u

v

H � 1

b1

1CCCA ; (14)

~R3 =

0BBB@

0B

�A�v

1CCCA ; ~R4 =

0BBB@

1u+Ac

v +Bc

H + uc

1CCCA ; (15)

where

q2 = u2 + v2; u = Au+Bv; v = Av �Bu; (16)

� =pe�; c =

qp� +

�p� ; H =

E + p

�; (17)

b1 =�

c2; b2 = 1 + b1q

2 � b1H: (18)

The eigensystem for the one-dimensional Euler equations are obtainedby setting v = 0.

2.2 Level Set Equation

We use the level set equation

�t + u�x + v�y = 0 (19)

to keep track of the interface location as the zero level of �. In general �starts out as the signed distance function, is advected by solving equation19 using the methods in [11], and then is reinitialized using

�t + S(�o)�q

�2x + �2y � 1�= 0 (20)

to keep � approximately equal to the distance function near the interfacewhere we need additional information. In this equation, S(�o) is the signfunction of �o with appropriate numerical smearing. More details are givenin the appendix.

7

We note that our method allows us to solve equation 19 independentlyof the Euler equations. That is, equation 19 can be solved directly usingthe method in [11], and the eigensystem for the Euler equations does notdepend on �, since we will be solving only one phase problems with any giveneigensystem (see the later sections). For details on the level set function see[24, 30].

2.3 Equations of State

We will use the following equations of state in our numerical examples.

2.3.1 Gamma Law gas

For an ideal gas p = �RT where R = Ru

M is the speci�c gas constant, withRu � 8:31451 J

molKthe universal gas constant and M the molecular weight

of the gas. Also valid for an ideal gas is cp � cv = R where cp is the speci�cheat at constant pressure. Additionally, gamma as the ratio of speci�c heats = cp

cv. [1]

For an ideal gas, equation 4 becomes

de = cvdT (21)

and assuming that cv does not depend on temperature (calorically perfectgas), we integrate to obtain

e = cvT (22)

where we have set e to be zero at 0K. Note that e is not uniquely determined,and we could choose any value for e at 0K (although one needs to usecaution when dealing with more than one material to be sure that integrationconstants are consistent with the heat release in any chemical reactions thatoccur).

Note that we may write

p = �RT =R

cv�e = ( � 1)�e (23)

for use in the eigensystem.

8

2.3.2 Tait Equation of State for Water

We use a sti� equation of state for the water,

p = B

��

�o

� �B +A (24)

where = 7:15, A = 105Pa, B = 3:31 � 108Pa, and �o = 1; 000 kgm3 . In

addition, we de�ne

e =B� �1

( � 1)� o+B �A

�(25)

at the internal energy per unit mass. [32]Note that this equation of state has pe = 0 which causes a division

by zero in the fourth component of ~R2. This can be avoiding with simplerescaling of ~L2 and ~R2 by dividing and multiplying by b1 respectively. Thenew eigenvectors become

~L2 =��q2 +H; u; v;�1� (26)

and

~R2 =

0BBB@

b1b1u

b1v

b1H � 1

1CCCA (27)

In addition, to model cavitation, the minimum pressure is set to bepmin = 22:0276Pa [32]. That is, the equation of state becomes p = pmin

for all densities that would admit pressures lower than pmin. Thus, all par-tial derivatives of pressure are identically zero for densities below

�min = �o

�pmin �A+B

B

� 1

(28)

and this causes problems in the eigensystem since the sound speed is nowzero. To remedy this problem we use a central scheme [22] when � < �min.

9

2.3.3 JWL Equation of state for Explosive Products

We use the following JWL (Jones-Wilkins-Lee) equation of state for explosiveproducts,

p = A

�1� !�

R1�o

�exp

��R1�o

�

�+B

�1� !�

R2�o

�exp

��R2�o

�

�+ !�e(29)

where A = 5:484 � 1011Pa, B = 9:375 � 109Pa, R1 = 4:94, R2 = 1:21,! = :28, and �o = 1; 630 kg

m3 . [32]

10

3 Numerical Method

We use the level set function to keep track of the interface. The zero levelmarks the location of the interface, while the positive values correspond toone uid and the negative values correspond to the other. Each uid satis�esthe Euler equations described in the last section with di�erent equations ofstate on each side. Based on the work in [11], the discretization of the levelset function can be done independent of the two sets of Euler equations.

Besides discretizing equation 19 we need to discretize two sets of Eulerequations. This will be done with the help of ghost cells. We will describethe scheme with an excessive use of ghost cells for the sake of clarity, andcomment on e�ciency later.

Given a level set function, it de�nes two separate domains for the twoseparate uids, i.e. each point corresponds to one uid or the other. Ourgoal is to de�ne a ghost cell at every point in the computational domain. Inthis way, each grid point will contain the mass, momentum, and energy forthe real uid that exists at that point (according to the sign of the level setfunction) and a ghost mass, momentum, and energy for the other uid thatdoes not really exist at the point (it is on the other side of the interface).Once the ghost cells are de�ned, we can use standard methods, e.g. see [29],to update the Euler equations at every grid point for both uids. Then weadvance the level set function to the next time step and use this to determinewhich of the two multidimensional spatial discretizations to use at a givengrid point. This makes multidimensional implementation trivial, since it isdone in the usual straightforward way, i.e. in the usual way for a single phase uid with no special concern for the interface, e.g. see [29]. In contrast, [10],[4] and [3] all need ill-advised dimensional splitting for multidimensionalproblems.

Consider a general time integrator for the Euler equations. In general,we construct right hand sides of the ordinary di�erential equation for both uids (based on the methods in [29]), then we advance the level set to thenext time level and pick one of the two right hand sides to use for the Eulerequations based on the sign of the level set function. This can be done forevery step and every combination of steps in a multistep method. Sinceboth uids are solved for at every grid point, we just choose the appropriate uid based on the sign of the level set function. This is incredibly simple to

11

program and apply as opposed to complexity and decision making involvedwith the use of multilevel time integrators in [10], [4] and [3].

To summarize, the method described here is trivial to implement. Useghost cells to de�ne each uid at every point in the computational domain.Update each uid separately in multidimensional space for one time stepor one substep of a multistep time integrator with standard methods. Thenupdate the level set function independently using the real uid velocities andthe sign of the level set function to decide which of the two answers is thevalid answer at each grid point. Keep the valid answer and discard the otherso that only one uid is de�ned at each grid point. Note that multistep timeintegrators will also require one to save the right hand side of the ordinarydi�erential equation for both uids for possible use at a later time level.Then de�ne new ghost cells and start over. In this we have regulated all thedi�cult decision making about special cases on interface crossing, cut cells,etc. to the subroutine that decides how to de�ne the ghost cells. In fact,the entire method relies on the ability to produce ghost cells that satisfy theappropriate boundary conditions for the Euler equations. In this way, onecan compute solutions to multiphase ow problems with one's own favoritesingle phase solver by adding a new routine to de�ne and deal with ghostcells. We chose to use the ENO scheme and TVD Runge Kutta methodsfrom [29].

Lastly, we note that only a band of 3 to 5 ghost cells on each side ofthe interface is actually needed by the computational method dependingon the stencil and movement of the interface. One can optimize the codeaccordingly.

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4 De�ning the Ghost Cells

Since a standard one phase solver will be used, the ghost nodes are thekey to the numerical method. We have discovered that a straightforwardboundary condition capturing approach yields surprisingly good results as isdemonstrated by our numerical examples.

4.1 One Dimension

To de�ne the ghost nodes in one spatial dimension, three quantities mustbe de�ned in the ghost region, then the equation of state along with theappropriate algebraic relations can be used to get the mass, momentum, andenergy.

We choose pressure and velocity as two of our three variables for physicalreasons. In many problems, pressure and velocity are continuous acrossthe interface and we can set the pressure and velocity of the ghost uididentically equal to the pressure and velocity of the real uid at each point.That is, node by node we can copy the real uid values of pressure andvelocity into the ghost uid values of pressure and velocity. In this waywe capture the interface boundary conditions for the pressure and velocitywithout explicitly identifying the interface location. Some modi�cation ofthis procedure is necessary when the pressure and velocity are discontinuousas will be discussed in a future paper.

Once the pressure and velocity have been de�ned at each ghost node, onemore quantity needs to be de�ned. In [6], it was shown that one degree offreedom exists at a material interface or contact discontinuity. This degreeof freedom corresponds to the advection of entropy in the linearly degenerate�eld. Note that entropy is generally discontinuous at a contact discontinuity.When one applies a standard �nite di�erence scheme to a discontinuousfunction, large errors result since the truncation error is not small. Shockcapturing methods have traditionally avoided the large dispersive errors witha myriad of special techniques while still allowing the large dissipative errorsthat are usually harmless in a one phase computation. However, these largedissipative errors can be the source of spurious oscillations in a two phasecomputation.

13

We eliminate the dissipative errors in the numerical method by usingone sided extrapolation of the entropy. De�ning the ghost cells with onesided extrapolation of the entropy will create a continuous entropy pro�leand remove the large errors due to numerical dissipation. Note that thediscontinuous nature of the entropy pro�le dictates that one sided extrapo-lation will capture the appropriate boundary condition. As discussed in [6],there is a true degree of freedom at a contact discontinuity and one has somechoice as to which variable to extrapolate, although one needs to use cautionsince there will be di�erent degrees of \overheating" errors depending on thevariable chosen. See [6] for details.

At this point, we describe the method in detail. Suppose that the zerolevel of the level set function lies between nodes i and i + 1, i.e. the levelset function changes sign between these nodes. Then uid 1 is de�ned atnode i and to the left of node i, while uid 2 is de�ned at node i+ 1 and tothe right of node i + 1. In order to update uid 1, we need to de�ne ghost uid values of uid 1 at nodes to the right and including node i + 1. Foreach of these nodes, we de�ne the ghost uid value by combining uid 2'spressure and velocity at each node with the entropy of uid 1 from nodei. This is constant extrapolation of entropy which is actually preferred overhigh order extrapolation since our interface will behave in a fashion similarto the piston in [6] su�ering from \overheating" errors. In fact we alwaysuse constant extrapolation of entropy to minimize the \overheating" errors.Likewise, we create a ghost uid for uid 2 in the region to the left andincluding node i. This is done by combining uid 1's pressure and velocityat each node with the entropy of uid 2 from node i+ 1.

As discussed in [6], the isobaric �x technique can be used to reduce the\overheating" errors. This technique allows the entropy in real uid values tochange. In order to apply our isobaric �x technique, we change the entropyat node i to be equal to the entropy at node i � 1 without modifying thevalues of the pressure and velocity at node i. Likewise, we change the entropyat node i + 1 to be equal to the entropy at node i + 2. This completes theisobaric �x, and then the ghost cells are de�ned as outlined above using thesenew values for the entropy.

Note that the isobaric �x can be combined with ghost node population ina simple way. For the nodes to the right and including node i, combine thepressure and velocity of each node with the entropy from node i � 1. Thisde�nes uid 1 to the right and including node i. For the nodes to the left andincluding node i+1, combine the pressure and velocity of each node with theentropy of node i+2. This de�nes uid 2 to the left and including node i+1.

14

This method is especially e�ective in multidimensional implementation.An important aspect of this method is its simplicity. We do not need to

solve a Riemann problem, consider the Rankine-Hugoniot jump conditions,or solve an initial boundary value problem at the interface. We capture theappropriate interface conditions by de�ning a uid that has the pressureand velocity of the real uid at each point, but the entropy of some other uid. Consider the case of air and water. In order to solve for the air,we replace the water with ghost air that acts like the water in every way(pressure and velocity) but appears to be air (entropy). In order to solvefor the water, we replace the air with ghost water that acts like the air inevery way (pressure and velocity) but appears to be water (entropy). Sincethe ghost uids behave in a fashion consistent with the real uids that theyare replacing, the appropriate boundary conditions are captured. Since theghost uids have the same entropy as the real uid that is not replaced, weare solving a one phase problem. We name this method the "Ghost FluidMethod", not to be confused with ghost cells or ghost nodes which are usedin the implementation of the method and have been in use for quite sometime.

4.2 Justi�cation

Here we provide a justi�cation of why our method works. Consider the caseof a solid wall boundary, where a re ection condition is used for the ghostcells. One can think of this as prescribing waves in the ghost region whichare identical to those in the real uid so that the real uid does not escapewhen it interacts with the boundary. Instead, it sees its re ected twin andbehaves as if the boundary was impenetrable [33]. Now consider an interfaceanywhere in a uid. We want the uid on one side of the interface to behavein the appropriate way when we add our ghost cells, and thus the easiestthing to do is to let all the ghost values be equal to the real uid values atthat point. In this way, the ghost cells do nothing and the scheme is just thestandard Eulerian scheme.

Unfortunately this standard Eulerian scheme does not behave well incertain situations, just as the piston does not behave well in [6] due to \over-heating". This implies that a simple modi�cation of the ghost cells is neededsimilar to [6]. We noticed in [6], that the only modi�cation necessary to cure\overheating" was an isobaric �x. If one thinks of the smearing out of thedensity pro�le in a contact discontinuity as a phenomena similar to \over-

15

heating" than it becomes obvious that the isobaric �x technique will workwell here. Thus, the only modi�cation in the ghost cells is to use the isobaric�x technique, while leaving the pressure and velocity unchanged.

4.3 Multidimensions

The above procedure is for one dimensional problems and as trivial as itseems, it does the job. For multidimensional problems, the ghost cells be-come more involved. We have more than one velocity to deal with, and weneed to make some choices for the direction of extrapolation. In multidi-mensions, we treat the pressure and the three dimensional velocity �eld inthe usual way, just de�ning the ghost values equal to the real uids values.In order to �nish the ghost cell procedure, we need to apply the isobaric�x technique to all the cells bordering the interface, and we need to extendthe isobaric �x variable into the ghost region in a fashion that resembles theconstant extrapolation done in the one dimensional case.

A natural way applying the isobaric �x technique exists because of thelevel set formulation. Using the level set function, we can de�ne the unitnormal at every grid point as

~N =~5�j~5�j

(30)

and then solve a partial di�erential equation for constant extrapolation inthe normal direction. This equation is

It � ~N � ~5I = 0 (31)

where I is the isobaric �x variable, e.g. the entropy. Note that the normal,~N , always points from the negative uid into the positive uid. We use the\+" sign in equation 31 to populate a ghost uid in the region where � > 0with the values of I from the region where � < 0, while keeping the real uidvalues of I �xed in the region where � < 0. Likewise, we use the \-" signin equation 31 to populate a ghost uid in the region where � < 0 with thevalues of I from the region where � > 0, while keeping the real uid valuesof I �xed in the region where � > 0. This equation only needs to be solvedfor a few time steps to populate a thin band of ghost cells needed for thenumerical method. Once the ghost cells are populated we can reassemblethe conserved variables.

16

Note that the above procedure does not apply an isobaric �x to the cellsin the real uid which border the interface. In order to apply the isobaric�x, we keep the real uid values of I �xed in the region where � < �� whenusing the \+" sign in equation 31, and we keep the real uid values of I �xedin the region where � > � when using the \-" sign in equation 31. Since �is an approximate distance function, we choose � to be the thickness of theband in which we wish to apply the isobaric �x. We use � = 1:54x.

4.4 Boundary Conditions

In some multimaterial problems, large jumps in tangential velocity exist atthe interface similar to the jumps in density and equation of state that weremedy in this paper. Most schemes will smear out this jump in tangentialvelocity due to numerical dissipation. An extension of our method allowsone to avoid this smearing.

We use the interface normal, ~N , to separate the three component velocity�eld into a tangential and a normal component. The normal component istreated in the same fashion as the velocity in one dimension, i.e. we copy thenormal velocity directly into the ghost cells with no change. The tangentialvelocity is handled in the same way as the isobaric �x variable, i.e. the goalis to extrapolate it or extend it as a constant in the normal direction. In twodimensions, a tangent vector must be chosen consistently in one direction orthe other. In three dimensions, one has a di�cult time choosing a consistenttwo dimensional basis for the tangent plane. We remove the di�culty inextension to higher dimensions by applying a basis free projection methodsimilar to the CPM (Complementary Projection Method) [8].

We de�ne the normal at each point by equation 30 and the velocity as~V =< u; v; w >. Once these are de�ned, we solve the propagation equation31 where ~I is now a column vector of length four which contains the threedimensional velocity �eld and the isobaric �x variable. Then at every cellin the ghost region we have two separate velocity �elds, one from the real uid and one from the ghost uid. Then for each velocity �eld, the normalcomponent of velocity, VN = ~V � ~N , is put into a three component vector,VN ~N , and then we use a complementary projection idea to de�ne the twodimensional velocity �eld in the tangent plane by another three componentvector, ~V � VN ~N . Then we take the normal component of velocity, VN ~N ,from the real uid and the tangential component of velocity, ~V �VN ~N , fromthe ghost uid and add them back together to get our new velocity to occupy

17

the ghost cell.We present a simple model to illustrate how the method works. Consider

a line in two dimensions or a plane in three dimensions which de�nes amaterial interface. Assume that the normal component of velocity is constantacross the interface, while the tangential velocity is constant on each side ofthe interface but jumps across the interface. Consider populating one of theghost regions with the velocity from the other side of the interface. Sincethe entire velocity �eld on one side of the interface is a constant, we arejust advecting that constant value into the ghost region. Then we split thevelocity �eld into a normal and tangential component for both the ghostcells and the real uid. We keep the tangential component from the ghost uid and the normal component from the real uid. Since the real uid hasthe same normal velocity on both sides of the interface, our procedure isequivalent to just keeping both components of the ghost cell velocity �eld.This is equivalent to using a constant velocity �eld, and our method has noknowledge of a jump in velocity at the interface. This allows our methodto completely avoid smearing and leads to exact modeling of planar shearwaves.

Shear waves may or may not be stable [23]. For example, shear waves arestable in high Mach number ows and when materials have strength (suchas steel). Besides the obvious smearing errors, standard schemes may su�erother problems due to their inability to correctly model these shear waves.For example, a shear wave moving across the grid will su�er from a pressureovershoot, while our scheme does not have this problem. In addition, thereare many large forces that may be incorrectly excited in material models dueto erroneous smeared out velocity pro�les. For example, consider two piecesof steel slowly sliding past each other at room temperature. The velocitypro�le will smear, inducing a continuous, non-constant velocity pro�le ineach piece of steel. This erroneous non-constant velocity pro�le will inducelarge non-physical resistant forces from a continuum model.

In many cases a jump in tangential velocity is not stable, and will leadto a Kelvin-Helmholtz instability. This instability is not well posed for theEuler equations, and only becomes well posed when viscosity (or some otherregularization such as surface tension) is added, e.g. Navier-Stokes ow.

18

4.5 A Note On Conservation and Convergence

Here, we brie y discuss the issues of conservation and convergence of thenumerical algorithm. The method described in this paper breaks the com-putational domain into two separate uids. Within each uid a standardconservative ux di�erencing scheme is used. At the interface between thetwo uids, there is formally a lack of discrete conservation on a set of mea-sure zero. Fluxes that exists between two di�erent uids are not unique.Since the pressure and normal velocity in each uid are the same, these uxes do have a unique pressure and normal velocity. However, they di�erin their values of entropy and tangential velocities. We note that entropyand tangential velocities move with the speed of the uid, so they do notcross the interface. In addition, there will be a lack of conservation due tothe advection of the level set function, �, similar to the area loss problemseen in incompressible ow calculations [30].

Since the scheme is formally non-conservative at the interface, we expectour scheme to behave like a fully conservative scheme with an O(�xn) sourceterm acting over the material interface. Here n is related to the order atwhich we are specifying the ghost node states and the order in which we im-plement the level set method. If the interface length, L(t), is independent ofresolution, then the overall lack of conservation will be of O(�xn

R t0L(t)dt).

Clearly if n > 0, one will achieve conservation. See Section 5.7 for an exam-ple where it appears that n = 2. Since in this case, conservation is achievedunder resolution, and since our discretization is numerically consistent withthe Euler equations, we expect to also get convergence to the proper weaksolution. Again this is seen in Section 5.7.

In general, we expect that for stable interface ows, the above argumentswill hold, and the algorithm should achieve both convergence and conserva-tion under mesh re�nement. Unfortunately, the inviscid Euler equations willgenerally be unstable at material interfaces due to either Kelvin-Helmholtz[12] or Richtmyer-Meshkov [27] types of instabilities. In these cases, thegrowth rate of an in�nitesimal disturbance is usually proportional to thewavenumber of the disturbance [23] and L(t) is resolution dependent. Sinceunder re�nement �ner scales are introduced, it is likely that L(t) / 1=�xm.Here, it is most likely thatm > 0, and the length of the interface will becomelarger under re�nement. In this case the error in conservation would be ofO(�xn�m

R t0L�(t)dt), where L� is the length of the interface at a particular

resolution (i.e. �xed). The fact that the interface may be growing is broughtoutside the integral and is grouped with n. We expect conservation under

19

mesh re�nement when n > m, and expect to lose conservation when n < m.Note that it is very di�cult in general to determine both n and m just givensome initial/boundary value problem. It may be possible that for a physi-cally unstable problem that n =m, in which case under re�nement one mayobserve a �xed (and possibly small) error in conservation. The example inSection 5.8 may be of this type, but it is probably best to simply monitor theerror in conservation for each problem and to attempt to determine n �m

numerically.It should be noted that many \fully conservative" schemes may conserve

overall mass, but may not conserve mass of each constituent [24]. In addi-tion, problems where the Euler equations have instabilities at all wavelengthswill never be a resolved even with a perfectly conservative scheme. For themethod described in this paper, conservation is achieved under resolutionfor problems that have a resolvable solution.

20

5 Examples

Unless otherwise noted the calculations are done with 3rd order ENO-LLF(essentially non-oscillatory - local Lax-Friedrichs) and 3rd order TVD RK(total variation diminishing Runge-Kutta) [29], except where the water cav-itates where the 3rd order central scheme [22] is used for the spatial dis-cretization.

5.1 Example 1

In this �rst example, we explore a simple one phase problem where an Eu-lerian scheme works well with no oscillations. We will compare our schemeto the standard Eulerian scheme.

This problem was taken from [32]. Consider a gamma law gas with = 1:4 on a 4m domain with 100 grid points. The interface is locatedmidway between the 50th and 51st grid points with left and right statesde�ned as �L = 2 kg

m3 , �R = 1 kgm3 , pL = 9:8� 105Pa, pR = 2:45� 105Pa, and

uL = uR = 0ms . We ran the code to a �nal time of .0022 seconds.The results in �gure 1 were obtained with the standard scheme while the

results in �gure 2 were obtained with the use of the new ghost cell techniquewhere we choose the isobaric �x variable to be entropy for extrapolation,but do not use the isobaric �x itself. Notice the slight overheating near thecontact discontinuity. In �gure 3, we add the constant entropy isobaric �xto clean up the overheating. All three sets of results are plotted on top ofthe exact solution.

Note that we still capture the shock and still generate the large dissipa-tive errors characteristic of shock capturing schemes. However, our contactdiscontinuity no longer su�ers from this dilemma.

5.2 Example 2

In this example we compute solutions to \Test A", \Test B", \Test C", andthe two cases of \Test D" from [17] where we have redimensionalized theproblems. Note that all of these examples have solutions where the pressureis constant across the contact discontinuity. Because of this, the pressure

21

evolution equation gives good results which are also shown in [17], althoughthere are large smearing errors due to numerical dissipation.

5.2.1 \Test A"

For \Test A" we use a 1m domain with 100 grid points. The interface islocated midway between the 50th and 51st grid points with left and rightstates de�ned as L = 1:4, R = 1:2, �L = 1 kg

m3 , �R = :125 kgm3 , pL = 1�105Pa,

pR = 1 � 104Pa, and uL = uR = 0ms . We ran the code to a �nal time of.0007 seconds and the results with the standard scheme from [24] are shownin �gure 4, while the results using our new scheme with entropy as theisobaric �x variable are shown in �gure 5. Both sets of results are plottedon top of the exact solution.

Note that the contact discontinuity is shifted one grid point to the left,since we estimate its speed with the local uid velocity when advecting thelevel set function. During wave interactions, the actual velocity of a contactdiscontinuity can vary slightly from the local uid velocity. We have per-formed a grid re�nement analysis and the contact discontinuity seems to beo� by one grid cell for all levels of grid re�nement yielding �rst order conver-gence in location as expected for a discontinuity where exact conservation isrelaxed slightly. A more resolved solution with 400 grid points is shown in�gure 6.

5.2.2 \Test B"

For \Test B" we use a 1m domain with 100 grid points. A right going shockis located midway between the 5th and 6th grid points and an interface islocated midway between the 50th and 51st grid points. The left, middle,and right states are de�ned as L = 1:4, M = 1:4, R = 1:2, �L = 1:3333 kgm3 ,�M = 1 kg

m3 , �R = :1379 kgm3 , pL = 1:5�105Pa, pM = 1�105Pa, pR = 1�105Pa,

uL = :3535p105ms , and uM = uR = 0ms . We ran the code to a �nal time

of .0012 seconds and the results with the standard scheme from [24] areshown in �gure 7, while the results using our new scheme with entropy asthe isobaric �x variable are shown in �gure 8. Both sets of results are plottedon top of the exact solution.

In this case, the contact discontinuity is located in the correct cell. Notethat the weak rarefaction wave (located to the left) and the weak shock wave

22

(located to the right) both su�er from numerical dissipation at this level ofresolution, independent of the sharp contact discontinuity. A more resolvedsolution with 400 grid points is shown in �gure 9.

5.2.3 \Test D", Case 1

This is similar to \Test B", except that we increase the strength of the shockwith �L = 4:3333 kg

m3 , pL = 1:5 � 106Pa, and uL = 3:2817p105m

s. We ran

the code to a �nal time of .0005 seconds and the results with the standardscheme from [24] are shown in �gure 10, while the results using our newscheme with entropy as the isobaric �x variable are shown in �gure 11. Bothsets of results are plotted on top of the exact solution.

In this case, the contact discontinuity is located in the correct cell. Notethat the glitch near x = :2m is due to the capturing of perfect shock initialdata by a shock capturing scheme. This is more pronounced in this example,since the shock wave is quite strong. If one starts with a smoothed out shockpro�le, this glitch is no longer present. A more resolved solution with 400grid points is shown in �gure 12.

5.2.4 \Test C"

This is similar to \Test B", except that we change the uid on the right to R = 1:249, �R = 3:1538 kg

m3 , pR = 1�105Pa, and uR = 0ms . We ran the codeto a �nal time of .0017 seconds and the results with the standard schemefrom [24] are shown in �gure 13, while the results using our new schemewith entropy as the isobaric �x variable are shown in �gure 14. Both sets ofresults are plotted on top of the exact solution.

In this case is located in the correct cell, although the shock wave locatedto the left is shifted two grid points to the right. Once again, these errorsare consistent under grid re�nement yielding �rst order accuracy in location.In addition, note that these errors do not increase in time, since they arethe result of estimating the velocity of the contact discontinuity by the local uid velocity during wave interactions. A more resolved solution with 400grid points is shown in �gure 15.

23

5.2.5 \Test D", Case 2

This is similar to \Test C", except that we increase the strength of the shockwith �L = 4:3333 kg

m3 , pL = 1:5 � 106Pa, and uL = 3:2817p105ms . We ran

the code to a �nal time of .0007 seconds and the results with the standardscheme from [24] are shown in �gure 16, while the results using our newscheme with entropy as the isobaric �x variable are shown in �gure 17. Bothsets of results are plotted on top of the exact solution.

In this case is located in the correct cell, although the shock wave locatedto the left is shifted two grid points to the right and the shock wave locatedto the right is shifted one grid point to the right. Note that the glitchesnear x = :3m and x = :7m are due to the capturing of perfect shock initialdata by a shock capturing scheme. If one starts with a smoothed out shockpro�le, these glitches are no longer present. A more resolved solution with400 grid points is shown in �gure 18.

5.3 Example 3

We take the initial data for the shock tube problem from \Test A" in [17]as in Example 2. This time we compute in two spatial dimensions on a 200by 200 grid with the shock tube aligned in the diagonal direction. In �gure19 we show output from the o�-diagonal direction. Note that we ran thecode for

p2 times longer in order to get a good comparison with \Test A"

in Example 2. The results are plotted on top of the exact solution.

5.4 Example 4

This problem was taken from [32]. Consider a 4m domain with 100 gridpoints and the interface located midway between the 50th and 51st gridpoints. There is a JWL gas on the left and water on the right with initialstates of �L = 1630 kg

m3 , �R = 1000 kgm3 , pL = 7:81� 109Pa, pR = 1:0� 105Pa,

and uL = uR = 0ms .Since the equation of state for water has pressure as a function of density

only, one needs to be careful when choosing the isobaric �x variable. Themost natural choice for water is the internal energy. For simplicity, we donot use the isobaric �x technique for the JWL gas, and we extend densitydirectly into the ghost cells.

24

We ran the code to a �nal time of .0005 seconds and the results usingour new scheme are shown in �gure 20. The results compare well with theexact solution in [32].

5.5 Example 5

This problem was taken from [32]. Consider a 10m domain with 500 gridpoints and re ection boundary conditions applied to both sides of the do-main. The interface is located midway between the 250th and 251st gridpoints, with a gamma law gas, = 1:25, on the left and water on theright. The initial states are �L = 1 kg

m3 , �R = 1000 kgm3 , pL = 1:0 � 105Pa,

pR = 1:0 � 105Pa, and uL = uR = 0ms . In addition, we have a shock ineach uid. Grid points 1 to 48 have � = 8:26605505 kg

m3 , p = 1:0 � 107Pa,and u = 2949:97131ms . Grid points 481 to 500 have � = 1004:1303 kgm3 ,p = 1:0� 107Pa, and u = �6:3813588m

s.

Since the equation of state for water has pressure as a function of densityonly, one needs to be careful when choosing the isobaric �x variable. Themost natural choice for water is the internal energy. We use entropy as theisobaric �x variable in the gas.

We ran the code to a �nal time of .003 seconds and the results usingour new scheme are shown in �gure 21 where we plot log10 � instead of thedensity, so that one may see the shock in the gas. In the �gure, we use REDfor the gas and GREEN for the water. In addition, note that the entropy�eld in the water is not used, so we set it to zero for graphing purposes. Theresults compare well with the solution computed in [32].

Note that the pressure evolution equation method in [17] has a di�culttime dealing with these sorts of contact discontinuities where the velocityand pressure are not both constant.

5.6 Example 6

This problem was taken from [32]. Consider a 10m domain with 400 gridpoints. A re ection boundary condition is applied to the left hand side of thedomain, while an out ow boundary condition applied to the right hand sideof the domain. Water is located in the central part of the domain surroundedby a gamma law gas, = 1:3, on each side. There are two interfaces, onebetween the 40th and 41st cell and one in between the 120th and 121st cell.

25

The gas has initial values of � = 35 kgm3 , p = 1:0 � 107Pa, and u = 500m

s,

while the water has initial values of � = 1004:1303 kgm3 , p = 1:0� 107Pa, andu = 500m

s.

This problem is computationally challenging so we modify our numeri-cal method slightly by choosing the high order viscosity for the ENO-LLFscheme as the largest of the three separate �eld viscosities as opposed to theusual �eld by �eld choice. In addition, we only use the second order accurateversion of the spatial method in the water.

Since the equation of state for water has pressure as a function of densityonly, one needs to be careful when choosing the isobaric �x variable. Themost natural choice for water is the internal energy. We use entropy as theisobaric �x variable in the gas.

We ran the code to a �nal time of .007 seconds and the results usingour new scheme are shown in �gure 22 where we plot log10 � instead of thedensity, so that one may see the shock in the gas. In the �gure, we use REDfor the gas and GREEN for the water. In addition, note that the entropy�eld in the water is not used, so we set it to zero for graphing purposes. Theresults compare well with the solution computed in [32].

Note that the pressure evolution equation method in [17] has a di�culttime dealing with these sorts of contact discontinuities where the velocityand pressure are not both constant.

5.7 Example 7

We examine the convergence and conservation of a stable ow �eld with aninterface. The problem is linear advection of a helium bubble in air and thenondimensionalized initial conditions are,

(� = 1; u = 1; v = 1; p = 1; = 1:4) air (32)

(� = :138; u= 1; v = 1; p = 1; = 1:67) helium (33)

� = �0:2 +q(x� :25)2+ (y � :25)2 level set (34)

where � � 0 represents helium and � > 0 represents the air. No reinitial-ization of the level set function was done. For this advection problem ourscheme achieves the exact state in each of the uid regions, and the only

26

error incurred is from the advection of the level set function. This may seemlike a trivial example, but most standard conservative or pressure evolutionschemes would smear out the density and possibly create spurious pressureoscillations.

A series of experiments were carried out on the unit square to measurethe convergence to the exact solution and to analyze how well the schemeconserves the mass of each uid. Zero gradient boundary conditions wereused for the conservative uid variables, and linear extrapolation at theboundaries was used for �. We used a centrally biased ENO scheme [28]applied in a central framework [22, 33] with third order TVD Runge-Kuttatime integration and third order WENO for the advection of � [15]. Weused �t = 0:1�x and integrated to t = 0:5. We measured two discreteerrors, namely the L1 error in the density �eld, E�, and the relative errorin total mass of helium, EHe, at t = 0:5. The errors and numerical rates ofconvergence, Rc, are given in Table I. Clearly the errors in both the density�eld and in total mass conservation converge at second order.

TABLE I: Numerical accuracy for helium advection in air.

�x = �y E� Rc EHe Rc

1/10 5:17� 10�2 5:00� 10�1

1/20 8:62� 10�3 2.58 8:16� 10�2 2.621/40 2:15� 10�3 2.00 2:03� 10�2 2.011/80 5:39� 10�4 2.00 5:02� 10�3 2.02

5.8 Example 8

In this example, we will illustrate the di�culty in computing shear waveswith shock capturing schemes and demonstrate the potential bene�ts of ournew method. A full computational analysis of these issues will be treated ina future paper on viscous ow.

Consider a 1m square domain with = 1:4, � = 1 kgm3 , and pL = 1�105Pa

everywhere. An interface is located at x = :5m with tangential velocities ofv = 300ms on the left and v = 200ms on the right. In addition, we impose anormal velocity of u = 50ms directed to the right. The ow is inviscid anda shear wave should be advected to the right with a perfect slip boundarycondition. We use a coarse mesh of 20 grid points in each direction and plot

27

the y = 10=19

mcross-section of this initial data in �gure 23. A shock capturing

scheme will smear out this shear wave creating the errors shown in �gure 24at .0005 seconds and �gure 25 at .005 seconds. Using the slip boundarycondition in our new scheme results in an extremely sharp solution as shownin �gure 26 at .005 seconds.

5.9 Example 9

We examine a Mach 1.22 air shock collapse of a helium bubble. Experimentalresults may be found in [13] and a numerical solution using adaptive meshre�nement (AMR) may be found in [31]. The physical initial conditions forthis problem are given in �gure 27, where the upper and lower boundary con-ditions are re ection for solid wall boundaries. The left and right boundaryconditions were zero gradient for the ow variables and linear extrapolationfor �. The nondimensionalized initial conditions are,

(� = 1; u = 0; v = 0; p = 1; = 1:4) pre-shocked air (35)

(� = 1:3764; u= �:394; v = 0; p = 1:5698; = 1:4) post-shocked air (36)

(� = :138; u= 0; v = 0; p = 1; = 1:67) helium (37)

� = �25 +q(x� 175)2+ y2 level set (38)

where � � 0 represents helium and � > 0 represents the air. The post-shockair state was given for x > 225. No reinitialization of the level set function orisobaric �x was done. We used a centrally biased ENO scheme [28] appliedin a central framework [22, 33] with third order TVD Runge-Kutta timeintegration and third order WENO for the advection of � [15]. Note thatthe computational domain was only the top half of the physical domain witha re ection condition applied at x = 0. A series of experiments were carriedout at di�erent resolutions (�x = 2; 1; 0:5; 0:25) at CFL = 0:8.

Figure 28 shows an idealized Schlieren image corresponding to 427�s afterthe air shock encounters the helium bubble (�x = 0:25). The image wasgenerated in the exact same manner as described in Section 3.3 of [31]. Alsoshown in �gure 28 is a circle representing the original helium-air interfaceto make the comparison easier with �gure 9(h) of [31] and �gure 7(h) of

28

[13]. The comparisons between the previous AMR solution and experimentare in good agreement for the general bubble shape and position. Thereare di�erences in the details of the interface, which is to be expected sincefor this problem the interface is unstable, and without some regularizationthere will be no unique or resolved answer to the Euler equations. For thisproblem the series of resolutions (�x = 2; 1; 0:5; 0:25) gave (2.5%, 0.78%,0.42%, 0.43%) as the time averaged relative percent errors in helium mass,respectively. Clearly this error in conservation of mass is not very signi�cant,and although it appears to be generally getting better with resolution, wemake no conjecture that conservation will be achieved under resolution tounstable problems.

29

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x 105 press

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

den

Figure 14: Test C - Ghost Fluid Method - 100 points

36

0 0.2 0.4 0.6 0.8 1

2

3

4

5

6

7

8

9

10

x 104 entropy

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

den

0 0.2 0.4 0.6 0.8 1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

x 105 press

0 0.2 0.4 0.6 0.8 1

0

20

40

60

80

100

120

vel

Figure 15: Test C - Ghost Fluid Method - 400 points

0 0.2 0.4 0.6 0.8 12

4

6

8

10

12

14

16

18

20

den

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

vel

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105 entropy

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

x 106 press

Figure 16: Test D, Case 2 - Standard Scheme - 100 points

37

0 0.2 0.4 0.6 0.8 12

4

6

8

10

12

14

16

18

20

22

den

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

vel

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105 entropy

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

x 106 press

Figure 17: Test D, Case 2 - Ghost Fluid Method - 100 points

0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14

16

18

20

22

den

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

1000

vel

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105 entropy

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

x 106 press

Figure 18: Test D, Case 2 - Ghost Fluid Method - 400 points

38

0 0.5 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

den

0 0.5 1−50

0

50

100

150

200

250

300

|vel|

0 0.5 1

1

1.05

1.1

1.15

1.2

1.25

x 105 entropy

0 0.5 1

1

2

3

4

5

6

7

8

9

10

x 104 press

Figure 19: Test A - 2D calculation - diagonal cross-section

0 1 2 3 4

1000

1100

1200

1300

1400

1500

1600

den

0 1 2 3 4

0

100

200

300

400

500

600

700

800

900

1000

vel

0 1 2 3 4

0

1

2

3

4

5

6

7

x 109 ene

0 1 2 3 4

0

1

2

3

4

5

6

7

8

x 109 press

Figure 20: JWL gas on the left & water on the right

39

0 2 4 6 8 10

−0.5

0

0.5

1

1.5

2

2.5

3

log(den)

0 2 4 6 8 10

−2500

−2000

−1500

−1000

−500

0

vel

0 2 4 6 8 10

0

2

4

6

8

10

12

14

16x 10

5 entropy

0 2 4 6 8 10

0

1

2

3

4

5

6

7x 10

7 press

Figure 21: gamma law gas (RED) & water (GREEN)

0 2 4 6 8 10

1

1.5

2

2.5

3

log(den)

0 2 4 6 8 10

0

100

200

300

400

500

vel

0 2 4 6 8 10

0

2

4

6

8

10

x 104 entropy

0 2 4 6 8 100

2

4

6

8

10

x 106 press

Figure 22: gamma law gas (RED) & water (GREEN)

40

0 0.2 0.4 0.6 0.8 10.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01den

0 0.2 0.4 0.6 0.8 149.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5uvel

0 0.2 0.4 0.6 0.8 1

−200

−100

0

100

200

300

vvel

0 0.2 0.4 0.6 0.8 10.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01x 10

5 press

Figure 23: Shear Wave - initial data

0 0.2 0.4 0.6 0.8 10.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

den

0 0.2 0.4 0.6 0.8 1

44

46

48

50

52

54

56

uvel

0 0.2 0.4 0.6 0.8 1

−200

−100

0

100

200

300

vvel

0 0.2 0.4 0.6 0.8 1

1

1.005

1.01

1.015

1.02

1.025

x 105 press

Figure 24: Shear Wave - Standard Scheme

41

0 0.2 0.4 0.6 0.8 10.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

den

0 0.2 0.4 0.6 0.8 1

48.5

49

49.5

50

50.5

51

uvel

0 0.2 0.4 0.6 0.8 1

−200

−100

0

100

200

300

vvel

0 0.2 0.4 0.6 0.8 1

0.99

0.995

1

1.005

1.01

1.015

x 105 press

Figure 25: Shear Wave - Standard Scheme

0 0.2 0.4 0.6 0.8 10.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01den

0 0.2 0.4 0.6 0.8 149.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5uvel

0 0.2 0.4 0.6 0.8 1

−200

−100

0

100

200

300

vvel

0 0.2 0.4 0.6 0.8 19.9

9.92

9.94

9.96

9.98

10

10.02

10.04

10.06

10.08

10.1x 10

4 press

Figure 26: Shear Wave - Ghost Fluid Method

42

Heliumbubble

Pre-shocked Air Post-shocked Air

(0mm,-44.5mm)

(325mm,44.5mm)

Figure 27: Physical Domain for Example 8

43

Figure 28: Schlieren image for Example 8 at t = 427�s (Rotated 90o clock-wise)

44

A Level Set Methods

In this section, we present some of the relevant ideas for discretization of thelevel set advection equation

�t + u�x + v�y = 0 (39)

and the reinitialization equation

�t + S(�o)�q

�2x + �2y � 1�= 0 (40)

We also discuss the advection equation for ghost cells population and theisobaric �x

It � nxIx � nyIy = 0 (41)

where we have used ~N =< nx; ny >.

A.1 Hamilton Jacobi Discretization

Following [25, 26], we need to �nd a left sided and right sided discretizationfor �x which we call ��x and �+x . The same procedure is applied to �y in theobvious fashion.

A.1.1 3rd Order ENO

We proceed along the lines of [29]. We will use polynomial interpolation to�nd � and then di�erentiate to get �x.

The zeroth order divided di�erences, D0i , and all higher order even di-

vided di�erences of � exist at the grid points and will have the subscript i.The �rst order divided di�erences, D1

i+ 1

2

, and all higher order odd divided

di�erences of � exist at the cell walls and will have the subscript i � 1

2.

Consider a speci�c grid point i0. To �nd ��x , set k = i0 � 1. To �nd �+x ,

set k = i0.De�ne

Q1(x) = (D1

k+ 1

2

�)(x� xi0): (42)

45

If jD2k�j � jD2

k+1�j, then c = D2k� and k? = k � 1. Otherwise, c = D2

k+1�

and k? = k. De�ne

Q2(x) = c(x� xk)(x� xk+1): (43)

If jD3

k?� 1

2

�j � jD3

k?+ 1

2

�j, then c? = D3

k?� 1

2

�. Otherwise, c? = D3

k?+ 1

2

�. De�ne

Q3(x) = c?(x� xk?)(x� xk?+1)(x� xk?+2): (44)

And then (��x )i0 is

D1

k+ 1

2

�+ c (2(i0 � k) � 1)4x+ c?�3(i0 � k?)2 � 6(i0 � k?) + 2

�(4x)2: (45)

A.1.2 5th Order WENO

We proceed along the lines of [16] and [15]. Consider a speci�c grid point i0.To �nd ��x , set

v1 =�i0�2 � �i0�3

4x ; v2 =�i0�1 � �i0�2

4x (46)

v3 =�i0 � �i0�1

4x ; v4 =�i0+1 � �i0

4x (47)

v5 =�i0+2 � �i0+1

4x (48)

and to �nd �+x , set

v1 =�i0+3 � �i0+2

4x ; v2 =�i0+2 � �i0+1

4x (49)

v3 =�i0+1 � �i0

4x ; v4 =�i0 � �i0�1

4x (50)

v5 =�i0�1 � �i0�2

4x (51)

46

Next we de�ne the smoothness

S1 =13

12(v1 � 2v2 + v3)

2 +1

4(v1 � 4v2 + 3v3)

2 (52)

S2 =13

12(v2 � 2v3 + v4)

2 +1

4(v2 � v4)

2 (53)

S3 =13

12(v3 � 2v4 + v5)

2 +1

4(3v3 � 4v4 + v5)

2 (54)

and the weights

a1 =1

10

1

(�+ S1)2; w1 =

a1a1 + a2 + a3

(55)

a2 =6

10

1

(�+ S2)2; w2 =

a2a1 + a2 + a3

(56)

a3 =3

10

1

(�+ S3)2; w3 =

a3a1 + a2 + a3

(57)

to �nally get (��x )i0 =

w1(v13� 7v2

6+11v36

) + w2(�v26

+5v36

+v43) + w3(

v33+5v46� v5

6) (58)

Note that we use � = 10�6.

A.2 Convection

In order to solve,

�t + u�x + v�y = 0 (59)

we look at the velocities. If ui0 > 0, then we use ��x . If ui0 < 0, thenwe use �+x . If ui0 = 0, then we do not need to choose either. The sameconsiderations apply to v and �y.

47

A.3 Reinitialization

In order to solve,

�t + S(�o)�q

�2x + �2y � 1�= 0 (60)

we can rewrite it as

�t +

S(�o)�xp�2x + �2y

!�x +

S(�o)�yp�2x + �2y

!�y = S(�o) (61)

and consider S(�o)�x and S(�o)�y evaluated at i0 to determine the upwinddirections. [30]

We use a modi�cation of Godunov's method [26]. If S(�o)�+x � 0 and

S(�o)��x � 0, then we use ��x . If S(�o)�

+x � 0 and S(�o)�

�x � 0, then we use

�+x . If S(�o)�+x > 0 and S(�o)�

�x < 0, then we use �x = 0. If S(�o)�

+x < 0

and S(�o)��x > 0, we de�ne

s =S(�o)(j�+x j � j��x j)

�+x � ��x(62)

and if s > 0, then we use ��x . Otherwise we use �+x .

The same procedure is repeated for S(�o)�y and the appropriate valuesfor �x and �y are plugged into equation 60.

Note that we smear out the sign function and de�ne

S(�o) =�p

�2 + (4x)2 (63)

instead of the exact sign function.We also use a limiter in the time evolution of the distance function to

keep the interface from crossing grid points.

A.4 Ghost Cells

In order to solve,

It + nxIx + nyIy = 0 (64)

we use a �rst order ENO approximation to I+x and I�x as outlined above for�. Note that we use �rst order since theoretically this equation is solved tosteady state.

48

We will also need to evaluate the unit normal

~N =< nx; ny >=

*�xp

�2x + �2y;

�yp�2x + �2y

+(65)

at i0, e.g. using central di�erencing. It is usually helpful to use a temporaryvariable, �, when computing the normal. First copy � into �, reinitialize � toproduce an approximate distance function, then compute the normals using�. In this way one can get improved values for the normal without losingaccuracy in the original level set function, �. Note that caution should beused to avoid division by zero when �x = �y = 0.

If nx > 0, then we use I�x . If nx < 0, then we use I+x . If nx = 0, thenwe do not need to choose either. The same considerations apply to the nyIyterm.

In order to solve,

It � nxIx � nyIy = 0 (66)

we use �nx instead of nx and �ny instead of ny in the procedure above.

We use + ~N to update the ghost cells with � > 0 and � ~N to update theghost cells with � � 0. where we have chosen the convention that � = 0belongs to the uid with � < 0. To apply the isobaric �x, we allow a band ofthe real uids cells near the interface to be populated along with the cells onthe other side of the interface. In this case, we use + ~N to update the cellswith � > �� and � ~N to update the cells with � < +� where � determinesthe thickness of the band. For example, choose � = 1:54x.

A.5 Time Evolution

The advection equation for the level set function is updated together withthe Euler equations.

The reinitialization equation is usually solved in �ctitious time after eachfully complete time step for the Euler equations. For example, set 4� = 4x

2

and take 10 � -steps with a 3rd order TVDRunge Kutta method to reinitializeabout 5 cells on each side of the interface to be approximately a distancefunction.

The advection equation for the population of ghost cells must be doneafter each substep of the time discretization for the Euler equations, in con-trast to the reinitialization. For example, the ghost cells must be populated

49

after each Euler substep of a Runge Kutta method, whereas the reinitializa-tion is done after each full Runge Kutta step. This update is also done in�ctitious time. For example, set 4� = 4x

2and take about 20 � -steps with a

3rd order TVD Runge Kutta method to populate a small band of ghost cells.We caution the reader that numerical dissipation could a�ect this ghost cellpopulation and that they may need more than 20 steps on occasion.

A.6 Boundaries

The following boundary condition keeps the characteristics owing outwardfor the level set function. After updating the interior of the domain, weupdate the boundary points with

�B = �B�1 + S(�B�1)j�B�1 � �B�2j (67)

where �B lies on the boundary and �B�1 and �B�2 are the adjacent pointsin a given coordinate direction.

50

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54